# MATHEMATICS 2 — SS 2017/18

Office hours:
• upon request (see my timetable and suggest me some days and times via e-mail)
• my office is in Sokolovská 83, 2nd floor, behind the glass door opposite to the staircase

## News

Exam D: problems and solutions and grading.

Exam C: problems and solutions and grading.

Exam B: problems and solutions and grading.

Exam A: problems and solutions and grading.

Complete Exam requirements and Sample question.

Sample test including solution and grading.

Script is available now (not final version).

## Basic information

• midterm test: 10 points
• homeworks: 20 points (10 homeworks, 2 points each)
• final exam: 90 points (50 points written exam, 40 points oral exam)
Students will be admitted to the oral exam only if the score from the first part (midterms, homeworks and written exam together) is at least 50 points. To pass the exam successfully, at least 20 points from the oral exam are required.

Grading: The total score is obtained as the sum of the points from the oral part and the first part, where the score of the first part is reduced to 60 if it exceeds 60. The final grade depends on the total score as follows.
• 70-75.5 points ... "E"
• 76-81.5 points ... "D"
• 82-87.5 points ... "C"
• 88-93.5 points ... "B"
• 94-100 points ... "A"

Midterm test takes part approximately in the half of the semester. Students are asked to

• find (and draw) the domain of a function of several variables and compute partial derivatives
within 30 minutes. Literature is allowed, electronic devices are prohibited. There are two attempts, the better result is taken into account. Here is
Midterm 1 and its solution, Midterm 2 and its solution.

Final Exam takes part in the examination period at the end of the semester. Students have three attempts to pass the final exam. It consists of a written part and oral part.
• Written part. Students have 90 minutes to solve problems on
• finding extrema of a function of several variables,
• matrix computations (system of linear equations, determinant, inverse matrix), and
• antiderivative (or Riemann integral).
Lecture notes and other materials are allowed, electronic devices are prohibited. A sample test is available here.
• Oral part follows typically the day after the written exam. The oral part tests understanding the definitions and theorems and ability to apply them. Each student should prepare answers within approximately 40 minutes. During the oral part only pencil and paper are allowed. Then the student should present answers and should answer additional questions. Here is a sample question.
See exam details for more details concerning both parts, as well as general exam rules and the requirements (list of required definitions, theorems and proofs).

## Lecture

Here is the beamer presentation to download and the list of definitions and theorems for printing (last update May 14, 2018).

Preliminary plan of lectures:
• lecture 1: Functions of several variables - analogies with functions of one variable, drawing graphs, computing partal derivatives
• lecture 2: the space Rn, open and closed sets in Rn, boundary, convergence of sequences in Rn,
• lecture 3: properties of open and closed sets, bounded sets
• lecture 4: continuous functions of several variables, Heine theorem, level sets of continuous functions
• lecture 5: compact sets in Rn, characterisation of compactness in Rn, extrema of continuous functions on compact sets
• lecture 6: partial derivatives, necessary condition for a local extremum, functions of class C1
• lecture 7: weak Lagrange theorem, tangent hyperplane, continuity of C1 functions
• lecture 8: chain rule, gradient, critical point, partial derivatives of higher order, finding extrema of functions
• lecture 9: implicit function theorem
• lecture 10: implicit function theorem
• lecture 11: Lagrange multipliers theorem
• lecture 12: finding extrema of functions
• lecture 13: concave functions, quasi-concave functions
• lecture 14: Matrix calculus - solving systems of linear equations, basic definitions
• lecture 15: matrix multiplication, matrix transpose, inverse matrix
• lecture 16: linear combination, transformation of a matrix
• lecture 17: determinant
• lecture 18: Gaussian elimination, Rouché-Fontené Theorem, Cramer's rule
• lecture 19: Antiderivative and Riemann integral - basic properties of antiderivatives, existence
• lecture 20: integration by parts, substitution
• lecture 22: polynomials and partial fractions
• lecture 24: Riemann integral - definition and basic properties
• lecture 25: Newton-Leibniz formula, substitution and integration by parts for the Riemann integral
• lecture +1: Applications of the Riemann integral
• lecture +2: (matrices as linear mappings)

## Seminar and exercises

Previous homeworks (without solutions)
HW1, HW2, HW3, HW4 (sketch of solution), HW5, HW6, HW7, HW8, HW9, HW10.

Problems to practice:
Preliminary plan:
• 1. week: partial derivatives
• 2. week: open and closed sets, cuts and contours
• 3. week: domains of functions of several variables, partial derivatives
• 4. week: extrema of functions
• 5. week: Lagrange multipliers
• 6. week: extrema of functions
• 7. week: implicit functions
• 8. week: systems of linear equations
• 9. week: inverse matrix, determinants
• 10. week: antiderivatives - substitutions, integration by parts
• 11. week: integration of rational functions (no seminar 1.5.)
• 12. week: Newton-Leibnitz formula (no seminar 8.5.)
• 13. week: Newton-Leibnitz formula (no exersises)
• suplementary week: Newton-Leibnitz formula